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Let a be a 2xx2 matrix with non-zero ent...

Let a be a `2xx2` matrix with non-zero entries and let `A^(2)=I`, where `I` is a `2xx2` identity matrix. Define Tr(A)= sum of diagonal elements of A and |A| = determinant of matrix A.
Statement 1 : Tr `(A) = 0`
Statement 2 : `|A|=1`

A

Statement -1 is True, Statement -2 is true, Statement -2 is a correct explanation for Statement-1.

B

Statement-1 is True, Statement -2 is True, Statement -2 is not a correct explanation for Statement -1.

C

Statement -1 is True, Statement -2 is False.

D

Statement -1 is False, Statement -2 is True.

Text Solution

Verified by Experts

The correct Answer is:
C
Let `{:A=[(a,b),(c,d)]:}a,b,c,dne 0`. Then,
`{:A=[(a,b),(c,d)][(a,b),(c,d)]=[(a^2+bc,ab+bd),(ac+dc,bc+d^2)]:}`
`:. A^2=I`
`rArr a^2+bc=1,bc+d^2=1,ac+dc=0 andab+bd=0`
`rArr (a^2+bc)-(d^2+bc)=0,c(a+d)=0and (a+d)b=0`
`rArr a^2-b^2=0 and a+d=0 [:' b ne 0, c ne 0]`
`rarr a+d=0rArr Tr(A)=0`
So, statement -1 true.
`absA={:[(a,b),(c,d)]:}=ad-bc=-a^2-bc [:'d=-a]`
`rArr absA=-1 [:' a^2+bc=1]`
So, statement -2 is not true.
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